Chapter 1 An Overview of Algebraic Curves and Cryptography
V. KUMAR MURTY
1.1 Introduction
1.2 The basic paradigm
1.3 The Diffie-Hellman decision problem
1.4 Constraints on the group
1.5 Abelian varieties over finite fields
1.6 Elliptic curves
1.7 Statistical results
1.8 Abelian varieties of higher dimension
1.9 Outline of contents
Chapter 2 School's Point Counting Algorithm
NICOLAS THERIAULT
2.1 Preliminaries
2.2 Division polynomials
2.3 Schoof's algorithm
2.4 Implementation
2.5 Improvements by Atkin and Elkies
2.6 Computing the modular equations
2.7 Computing Pl, 5 and
2.8 Computing the factor of fe
2.9 Parallelization
Chapter 3 Report on the Denef-Vercauteren/Kedlaya Algorithm
ZUBAIR ASHRAF ALI JUMA AND PRAMATHANATH SASTRY
3.1 Background
3.2 Generalities
3.3 Main strategy
3.4 Monsky-Washnitzer cohomology
3.5 Hyperelliptic curves
3.6 Data structures
3.7 Algorithm for lifting the curve to characteristic zero
3.8 Inversion
3.9 The 2-power Frobenius on K
3.10 The characteristic polynomial of Frobenius
3.11 Multiplication
3.12 Running times
3.13 Parallelization
Chapter 4 An Introduction to Gr5bner Bases
MOHAMMED RADI-BENJELLOUN
4.1 Introduction
4.2 GrSbner bases
Chapter 5 Cab Curves and Arithmetic on Their Jacobians
FARZALI IZADI
5.1 Introduction
5.2 Preliminaries
5.3 The Cab curves
5.4 Addition algorithm for Jacobian group in divisor representation
5.5 Addition algorithm for Jacobian group in ideal representation
Chapter 6 The Zeta Functions of Two Garcia-Stichtenoth Towers
KENNETH W. SHUM 6.1 Introduction
6.2 Background on zeta functions
6.3 The first Garcia-Stichtenoth tower
6.4 The second Garcia-Stichtenoth tower
6.5 Conclusion
Appendix: Counting points over P0 in GS1
Bibliography
Index
